For David did not ascend into the heavens, but he says himself:

‘The LORD said to my Lord,
“Sit at My right hand,
Till I make Your enemies Your footstool.”’

“Therefore let all the house of Israel know assuredly that God has made this Jesus, whom you crucified, both Lord and Christ.”

From my commentary:
In this Psalm, King David – by the inspiration of the Holy Spirit – recorded that Yahweh, Israel’s covenant God (The LORD), spoke to David’s Lord (my Lord) as God. Peter used this to show that the Messiah, who is the focus of Psalm 110, is in fact Divine – He is God.

Therefore let all the house of Israel know: The sermon concludes with a summary. Simply, all Israel should know that even though they crucified Jesus, God has declared Him both Lord and Christ.

Peter is in fact using the Old Testament to prove the Jesus was who He said he was.